collectedBasis_orthonormal — Mathlib

English

An instance of StarOrderedRing is established for endomorphisms over a complex Hilbert space.

Русский

Установлен инстанс StarOrderedRing для комплексно-гиильбертовых пространств.

LaTeX

$$StarOrderedRing (ContinuousLinearMap (id) H H)$$

Lean4

theorem collectedBasis_orthonormal [DecidableEq ι] {V : ι → Submodule 𝕜 E}
    (hV : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) (hV_sum : DirectSum.IsInternal fun i => V i)
    {α : ι → Type*} {v_family : ∀ i, Basis (α i) 𝕜 (V i)} (hv_family : ∀ i, Orthonormal 𝕜 (v_family i)) :
    Orthonormal 𝕜 (hV_sum.collectedBasis v_family) := by
  simpa only [hV_sum.collectedBasis_coe] using hV.orthonormal_sigma_orthonormal hv_family

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